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Theorem fusgrvtxfi 26666
 Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
Hypothesis
Ref Expression
isfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
fusgrvtxfi (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi
StepHypRef Expression
1 isfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 26665 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
32simprbi 492 1 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106  ‘cfv 6135  Fincfn 8241  Vtxcvtx 26344  USGraphcusgr 26498  FinUSGraphcfusgr 26663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-fusgr 26664 This theorem is referenced by:  fusgrfupgrfs  26678  nbfusgrlevtxm1  26725  nbfusgrlevtxm2  26726  nbusgrvtxm1  26727  uvtxnm1nbgr  26752  cusgrm1rusgr  26930  wlksnfi  27280  fusgrhashclwwlkn  27477  clwwlkndivn  27478  fusgreghash2wsp  27746  numclwwlk3lem2  27816  numclwwlk4  27818
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